arXiv:hep-th/0105086v2 29 Jul 2001
G¨ oteborg ITP preprint hep-th/0105086 May 10, 2001
GAMMA: A Mathematica package for performing Γ-matrix algebra and Fierz transformations in arbitrary dimensions Ulf Gran1
Department of Theoretical Physics Chalmers University of Technology and G¨ oteborg University SE-412 96 G¨ oteborg, Sweden
Abstract We have developed a Mathematica package capable of performing Γ-matrix algebra in arbitrary (integer) dimensions. As an application we can compute Fierz transformations.
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E-mail:
[email protected]
COPYRIGHT AND DISCLAIMER OF WARRANTY
c 2001 Ulf Gran. Copyright This documentation is part of the GAMMA package, available for download at http://fy.chalmers.se/~gran/, developed for Mathematica version 4.0. The GAMMA package is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY. Neither the author nor any distributor accepts responsibility to anyone for the consequences of using it or for whether it serves any particular purpose or works at all, unless he says so in writing. Permission is granted for use and non-profit distribution of this package, in its original form, providing that this notice be clearly maintained, but the right to distribute this package for profit or as part of any commercial product is specifically reserved for the author. The same permission and reservation applies to the formatted document distributed with this package. If you find the GAMMA package useful for doing research, acknowledging the author’s work by proper reference to the package would be appreciated.
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Introduction
The motivation to develop this package arose from calculations in eleven-dimensional supergravity [1, 2], which required an immense amount of both pure Γ-matrix algebra and Fierz transformations to be performed. The rule-based programming of Mathematica [3] is ideally suited for handling algebraic computations, compared to ordinary procedural programming. For a very nice comparison between rule-based and procedural programming, as well as a toy implementation of Γ-matrix algebra in four dimensions, see [4], which describes a Mathematica package performing Γmatrix algebra in (4−ǫ) dimensions. This was the first Mathematica implementation of Γ-matrix algebra according to the ’t Hooft-Veltman scheme. There are quite a few programs handling Γ-matrices in four-dimensions [5, 6, 7, 8, 9, 10, 11], intended to be used when computing Feynman diagrams. There is also a Mathematica package handling computations in D = 10, N = 1 supergravity, including Γ-matrix algebra, [12]. In section 2, we will introduce and explain the functions in the package concerning the algebraic manipulation of Γ-matrices. This will be done through simple examples. In section 3 we will turn to the functions concerning fierzing. A complete list of the functions in the package, including brief descriptions, can be found in appendix A. In appendix B we list the built-in Mathematica functions that are redefined by the package. GAMMA is available for download at http://fy.chalmers.se/~gran/.
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Γ-matrix algebra
To load the package type
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